我认为你的老师的答案是对的，要用 going。我的语感也是这样：英语不说 There won't be a lot of people to go. 这里应该是反射不定式的用法有关，但 people 和 to go 不能这样用。

There won't be a lot of people going there.【正确】

There won't be a lot of people to go there.【正确】

【说明】

（1）现在分词短语和动词不定式短语都可以作定语修饰存在句（即 there be）的主语。

（2）a lot of 通常用于肯定句，所以这两个句子中的 a lot of 最好用 many 替换。

我找到一个 there be sth to do 的句子：

但上面的句子与网友的句子有所不同：即网友的句子中 sb 为不定式的逻辑主语，但上面的句子不是。

没想到这样的问题会引起如此反响，看些例句吧：

1.The people that are doing illegal things in other countries - like begging - are already there. There won't be many to join them.(BBC)

2.There won't be enough people to work. There'll only be older people.(Forbes)

3.The trouble is, though, that unless something is done soon, there won't be enough people to work on these projects or those that might follow.（卫报）

对比用working：

A collapsing birth rate, people living longer and the attractiveness of the road south are all contributing to the looming demographic crisis in Scotland, where by 2020 there won't be enough people working to support those who are not.（卫报）

4.There will be someone to meet you at the airport.(Oxford Practice Grammar Advanced With key(2006)第102页）

5.There won’t always be an adoring mother to take the edge off the blows.（Oxford Dictionary of Current Idiomatic English I Verbs With Prepositions and Particles第326页）

如果问，这两种形式有什么区别，这倒是一个好问题，任何句法形式上的不同一般都会引起其意义的细微差异！

也是感觉going很顺。再比较了一下搜索结果，going完胜。

但说到具体原因，似乎又不那么清晰。尝试解释如下，只是抛砖引玉，不当之处，敬请谅解。

句子主干there won’t be已经是将来时态，后面单纯的定语go受此影响，不需要再用不定式表达将来的概念（不知道这是不是叫“辖域”。）。

类比下面的定语从句：

I will give you anything you ask for. （薄冰《高级英语语法》关于一般现在时用于表将来的从句部分）

貌似反射不定式以及带有情态意味等的不定式不受此限制。